Non-archimedean Yomdin-Gromov parametrizations and points of bounded height

We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C} ((t))$, in analogy to results by Pila and Wilkie, resp. by Bombieri and Pila. Along the way we prove, for definable functions in a general context of non-archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.Comment: 54 pages; revised, section 5.6 adde

Fonctions constructibles exponentielles, transformation de Fourier motivique et principe de transfert

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative cases. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We define also motivic Schwartz-Bruhat spaces on which motivic Fourier transformation induces an isomorphism. Our motivic integrals specialize to non archimedian integrals. We give a general transfer principle comparing identities between functions defined by integrals over local fields of characteristic zero, resp. positive, having the same residue field. Details of constructions and proofs will be given elsewhere.Comment: 10 page

Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero

We extend the formalism and results on motivic integration from ["Constructible motivic functions and motivic integration", Invent. Math., Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to previously known versions of motivic integration by e.g. the second author and J. Sebag and to classical p-adic integrals.Comment: 33 pages. Final versio

Energy Balance in the Solar Transition Region. IV. Hydrogen and Helium Mass Flows With Diffusion

In this paper we have extended our previous modeling of energy balance in the chromosphere-corona transition region to cases with particle and mass flows. The cases considered here are quasi-steady, and satisfy the momentum and energy balance equations in the transition region. We include in all equations the flow velocity terms and neglect the partial derivatives with respect to time. We present a complete and physically consistent formulation and method for solving the non-LTE and energy balance equations in these situations, including both particle diffusion and flows of H and He. Our results show quantitatively how mass flows affect the ionization and radiative losses of H and He, thereby affecting the structure and extent of the transition region. Also, our computations show that the H and He line profiles are greatly affected by flows. We find that line shifts are much less important than the changes in line intensity and central reversal due to the effects of flows. In this paper we use fixed conditions at the base of the transition region and in the chromosphere because our intent is to show the physical effects of flows and not to match any particular observations. However, we note that the profiles we compute can explain the range of observed high spectral and spatial resolution Lyman alpha profiles from the quiet Sun. We suggest that dedicated modeling of specific sequences of observations based on physically consistent methods like those presented here will substantially improve our understanding of the energy balance in the chromosphere and corona.Comment: 50 pages + 20 figures; submitted to ApJ 9/10/01; a version with higher resolution figures is available at http://cfa-www.harvard.edu/~avrett

Dimension dependence of correlation energies in two‐electron atoms

Correlation energies (CEs) for two‐electron atom ground states have been computed as a function of the dimensionality of space D. The classical limit D→∞ and hyperquantum limit D→1 are qualitatively different and especially easy to solve. In hydrogenic units, the CE for any two‐electron atom is found to be roughly 35% smaller than the real‐world value in the D→∞ limit, and about 70% larger in the D→1 limit. Between the limits the CE varies almost linearly in 1/D. Accurate approximations to real CEs may therefore be obtained by linear interpolation or extrapolation from the much more easily evaluated dimensional limits. We give two explicit procedures, each of which yields CEs accurate to about 1%; this is comparable to the best available configuration interaction calculations. Steps toward the generalization of these procedures to larger atoms are also discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70213/2/JCPSA6-86-6-3512-1.pd